In section 2.4.4 of Lurie's Higher Topos Theory, it is said multiple times that using

Proposition: Let $p : \mathcal{C} \rightarrow \mathcal{D}$ be an inner fibration of $\infty$-categories. Let $x,y$ be vertices of $\mathcal{C}$, let $\tilde{e} : p(x) \rightarrow p(y)$ be an edge of $\mathcal{D}$ and let $ e: x'\rightarrow y$ be a locally $p$-Cartesian edge of $\mathcal{C}$ lifting $\tilde{f}$. Then in the homotopy category $\mathcal{H}$ of spaces there is a fiber sequence
$$ Map_{\mathcal{C}_{p(x)}}(x,x') \rightarrow Map_{\mathcal{C}}(x,y) \rightarrow Map_{\mathcal{D}}(p(x),p(y))$$ where the fiber is taken over $\tilde{e}$.

we have that any cartesian fibration of simplicial set $p :\mathcal{C} \rightarrow \mathcal{D}$ is fully faithful map in the sense that for every vertices $x,y \in \mathcal{C}$, the induced map
$$Map_{\mathcal{C}}(x,y) \rightarrow Map_{\mathcal{D}}(p(x),p(y))$$ is a Kan weak equivalence of simplicial sets.

I see that from the fact that $p$ is a Cartesian fibration we have that $p$ satisfies the hypothesis of the above proposition. However I do not see how we can say that $p$ is fully faithful from it.

$\begingroup$He never makes that claim, and it’s not true (map a big simplicial set to a point). In each instance there are further properties assumed of p to ensure that fiber from 2.4.4.2 is zero, or the proposition is used in a different way. For example, in 2.4.4.4, we have a map between two loc Cartesian fibrations and it is assumed that the map preserves loc Cartesian edges and is a fiberwise equivalence- that tells you the map on fibers in 2.4.4.4 is an equivalence, and the map on bases is the identity, so the map in the middle is an equivalence.$\endgroup$
– Dylan WilsonJan 2 at 14:54

$\begingroup$Is there a specific use of 2.4.4.2 that you’re interested in?$\endgroup$
– Dylan WilsonJan 2 at 14:54

$\begingroup$I am mostly interested of its use in 2.4.4.5. I maybe should have written $\infty$-categories instead of simplicial sets.$\endgroup$
– Oscar P.Jan 2 at 15:04

4

$\begingroup$But it’s still false then- for example the map from Delta^1 to a point is a Cartesian fibration that is not fully faithful. In 2.4.4.5, the square with C’,C,D, and D’ induces a map of fiber sequences from 2.4.4.2. The map on bases is an equivalence bc D’—>D is assumed to be a categorical equivalence. The map on fibers is an isomorphism since C’ is the literal pullback. So the map in the middle is a weak equivalence, whence the claim about C’—>C being fully faithful.$\endgroup$
– Dylan WilsonJan 2 at 15:20

1

$\begingroup$That’s exactly the content of the two sentences I wrote preceding “So the map in the middle is a weak equivalence.”$\endgroup$
– Dylan WilsonJan 2 at 16:29